On the geometry of measures with density bounds in a H\"older anisotropic setting

Abstract

We study the regularity of the support of a Radon measure $\mu$ on $\mathbb R^{n+1}$ for which anisotropic versions of its $n$-dimensional density ratio and its doubling character are assumed to converge with H\"older rate. We show that in either case, if the support of $\mu$ is flat enough, then it is a $C^{1,\gamma}$ $n$-dimensional submanifold of $\mathbb R^{n+1}$, for some $\gamma\in (0,1)$. If the flatness assumption is dropped, then the support of $\mu$ is the union of a $C^{1,\gamma}$ $n$-dimensional submanifold of $\mathbb R^{n+1}$ and a closed singular set that is either empty if $n\leq 2$, or has Hausdorff dimension at most $n-3$ if $n\geq 3$.